In 2007, Simon [1]} proved that, given a family of ergodic Schrödinger operators \(H_\theta \)  with positive Lyapunov exponent, the spectral measure \(\mu _\theta \)  must be supported on a set with zero logarithmic capacity for a.e. \(\theta .\)  The ergodicity and positive Lyapunov exponent conditions in [1]} are more restrictive than our requiring positive upper Lyapunov exponent. Hence, Simon's result still leaves us with the question of what happens in the setting of positive upper Lyapunov exponent, as well as what happens on the excluded Lebesgue null set.
